Finite Group Databasebeta

← #45664508

Subgroups — #45664508

Order 32896.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551525C212yesmaximal, complemented
2C2×C22⋊#14523339C211yesmaximal, characteristic
3#30656597C211yesmaximal, characteristic
4#30656499C212yesmaximal, complemented
5C22×C4⋊C4×C257C211yesmaximal, characteristic
6C22×#14523339C2212yes—
7C2×C4⋊Dic257C2211yescharacteristic
8C2×C4⋊Dic257C2211yescharacteristic
9C2×C22⋊Dic257—21no—
10#22551525—24no—
11C22⋊#14523339C414yescomplemented
12#22551499—21no—
13C23×C4×C257—21no—
14C23×C4×C257C2211yescharacteristic
15C2×C4⋊C4×C257—21no—
16C2×C4⋊C4×C257—24no—
17C2×C4⋊C4×C257C2212yes—
18C22×Dic257—21no—
19C22×Dic257—21no—
20C2×#14523339—22no—
21C2×#14523339—22no—
22C2×#14523339C2×C414yescomplemented
23C4⋊Dic257—22no—
24C4⋊Dic257—22no—
25C22⋊Dic257—42no—
26C24×C257D411yescharacteristic
27C22×C4×C257—22no—
28C22×C4×C257—22no—
29C22×C4×C257—24no—
30C22×C4×C257—22no—
31C22×C4×C257—42no—
32C22×C4×C257—42no—
33C22×C4×C257C2×C412yes—
34C22×C4×C257D411yescharacteristic
35C22×C4×C257Q812yes—
36C22×C4×C257C2311yescharacteristic
37C4⋊C4×C257—24no—
38C4⋊C4×C257—22no—
39C4⋊C4×C257D414yes—
40C2×Dic257—41no—
41C2×Dic257—41no—
42C2×Dic257—41no—
43C2×Dic257—41no—
44C2×Dic257—41no—
45C2×Dic257—41no—
46#14523339—24no—
47C23×C257—21no—
48C23×C257—21no—
49C23×C257—21no—
50C23×C257—21no—
51C23×C257C4⋊C412yes—
52C23×C257D812yes—
53C23×C257Q1612yes—
54C23×C257C2×D411yescharacteristic
55C2×C4×C257—21no—
56C2×C4×C257—21no—
57C2×C4×C257—24no—
58C2×C4×C257—24no—
59C2×C4×C257—24no—
60C2×C4×C257—48no—
61C2×C4×C257—42no—
62C2×C4×C257—42no—
63C2×C4×C257C4⋊C412yes—
64C2×C4×C257C22×C411yescharacteristic
65C2×C4×C257C2×D412yes—
66C2×C4×C257C2×Q811yescharacteristic
67C2×C4×C257#4112yes—
68C22×C257—22no—
69C22×C257—22no—
70C22×C257—22no—
71C22×C257—22no—
72C22×C257—22no—
73C22×C257—22no—
74C22×C257#10714yescomplemented
75C22×C257C2×C4⋊C411yescharacteristic
76C22×C257#12011yescharacteristic
77C22×C257#12311yescharacteristic
78C22×C257C2×D811yescharacteristic
79C22×C257C2×Q1611yescharacteristic
80C22×C257AGL(1,ℤ/8)11yescharacteristic
81C22×C257#13711yescharacteristic
82C4×C257—22no—
83C4×C257—44no—
84C4×C257C4×D412yes—
85C4×C257C22⋊Q812yes—
86Dic257—81no—
87Dic257—81no—
88C2×C257—24no—
89C2×C257#38511yescharacteristic
90C2×C257C2×#10711yescharacteristic
91C2×C257#42811yescharacteristic
92C2×C257#44711yescharacteristic
93C2×C257#45111yescharacteristic
94C2×C257#48011yescharacteristic
95C2×C257#48411yescharacteristic
96C257#185211yescharacteristic, complemented, Sylow
97#1852—2571nomaximal, Sylow
98C22×C4⋊C4D25711yescharacteristic
99C2×#107—2572no—
100C2×C22⋊C8—2571no—
101#380—2571no—
102#340—2572no—
103C23×C4—21no—
104C23×C4C2×D25711yescharacteristic
105C22×C8—2572no—
106C2×C4⋊C4—2571no—
107C2×C4⋊C4—2571no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4C2×D25712yes—
111C2×C22⋊C4—5141no—
112#107—5144no—
113C22⋊C8—2574no—
114#95—5141no—
115C24C22⋊D25711yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—5141no—
121C22×C4—5141no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C4×D25712yes—
125C22×C4C22⋊D25711yescharacteristic
126C22×C4Dic51412yes—
127C22×C4C22×D25711yescharacteristic
128C2×C8—2574no—
129C2×C8—5142no—
130C2×C8—5142no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—5142no—
134C4⋊C4—5142no—
135C4⋊C4C22⋊D25714yes—
136C22⋊C4—10282no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23#1686159712yes—
142C23#1686160312yes—
143C23#1686160812yes—
144C23C2×C22⋊D25711yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—48no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—10281no—
154C2×C4—10281no—
155C2×C4—10281no—
156C2×C4—10281no—
157C2×C4—10281no—
158C2×C4—10281no—
159C2×C4#1686159712yes—
160C2×C4C2×C4×D25711yescharacteristic
161C2×C4C2×C22⋊D25712yes—
162C2×C4C2×Dic51411yescharacteristic
163C2×C4#1686161812yes—
164C8—5144no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#2255152514yescomplemented
172C22C2×#1686159711yescharacteristic
173C22#2255156111yescharacteristic
174C22#2255157511yescharacteristic
175C22C2×#1686160311yescharacteristic
176C22C2×#1686160811yescharacteristic
177C22#2255161811yescharacteristic
178C22#2255162311yescharacteristic
179C4—22no—
180C4—44no—
181C4—20561no—
182C4—20561no—
183C4C4×C22⋊D25712yes—
184C4#2255157112yes—
185C2—24no—
186C2#3065661011yescharacteristic
187C2C2×#2255152511yescharacteristic
188C2#3065677711yescharacteristic
189C2#3065687111yescharacteristic
190C2#3065689311yescharacteristic
191C2#3065705011yescharacteristic
192C2#3065707211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
FittingC22×C4⋊C4×C257C2
socleC23×C257C2×D4
prime radicalC24×C257D4
cosocleC22×C4×C257C23
hypercenterC22×C4⋊C4D257
hypocenterC257#1852
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22551525C2semidirect
4. #30656499C2semidirect
11. C22⋊#14523339C4semidirect
22. C2×#14523339C2×C4semidirect
74. C22×C257#107semidirect
96. C257#1852semidirect
171. C22#22551525semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225515252
C2×C22⋊#145233392
#306565972
#306564992
C22×C4⋊C4×C2572
#1852257